FISHERY BULLETIN: VOL. 79, NO. 4 



can be obtained from CPUE and effort data com- 

 piled from a unit stock fishery under equilibrium 

 conditions even w^hen the catchability coefficient 

 iq) is unknown. 



Since catch and CPUE will usually be related to 

 effort within a single year and during several pre- 

 ceding years, Gulland (1969) has argued that the 

 abundance of a particular year class which has 

 been in the fishery for jc years would be influenced 

 by the fishing mortalities during those x years and 

 proposed that the total CPUE for all ages during 

 any given year would be related to some weighted 

 mean of fishing effort in those jc years. According 

 to Gulland (1969), if this mean is taken over a 

 period of time equal to the mean duration of life in 

 the exploited phase, then the relation between 

 CPUE and this mean effort f will approximate 

 that between CPUE and effort in the steady 

 state. 



An exponential form of the surplus production 

 model was developed by Fox (1970) to express the 

 nonlinear relationship between CPUE and effort 

 statistics. The exponential surplus production 

 model was based on the assumption that the rate 

 of population increase was best described by the 

 Gompertz growth function rather than the logistic 

 growth function assumed by the linear form of the 

 model. For the exponential model, under equilib- 

 rium conditions 



dB 

 dt 



(10) 



and the annual equilibrium yield ( Ye) is therefore 



dB 

 dt 



Y^ = kB^{\og^B^-\og^B^). (11) 



Collecting terms and dividing through by Ue 



(=YElfE) 



or 



fE = -(»0g.f^.-l0ge^£) (14) 



\og,U^ = log,^^-!-)^^ 



(15) 



which is equivalent to 



Ur,. = U e -''fE 



(16) 



Multiplying Equation (16) by the fishing effort at 

 equilibrium (fg) 



Y,^ = UUe-^'E 



(17) 



Therefore, the effort ifs) which produces 

 maximum equilibrium yield (Y.s) is determined 

 from the relationship 



dYr^ 



dfi 



= -bf,M e-^tE+ U e-^'fE = (18) 



E 



and 



/■.,=-• 



(19) 



Since CPUE ( Y/f = U) is defined to be proportional 

 to population biomass 



Since bfs = 1, CPUE at fs (Us) is 



B, 



1 /Y 



Q\f, 



U, 



(12) 



U, = U e-bfs 



u 



(20) 



Equation (11) can be rewritten as 



and the maximum equilibrium yield (Ys) is given 

 by 



U 

 Y = f U = ^^ 

 ^ ' ' be 



(21) 



696 



